ON THE CONVERGENCE OF INEXACT ... - Semantic Scholar

TAIWANESE JOURNAL OF MATHEMATICS Vol. 18, No. 2, pp. 419-433, April 2014 DOI: 10.11650/tjm.18.2014.3066 This paper is available online at http://journal.taiwanmathsoc.org.tw

ON THE CONVERGENCE OF INEXACT PROXIMAL POINT ALGORITHM ON HADAMARD MANIFOLDS P. Ahmadi and H. Khatibzadeh* Abstract. In this paper we consider the proximal point algorithm to approximate a singularity of a multivalued monotone vector field on a Hadamard manifold. We study the convergence of the sequence generated by an inexact form of the algorithm. Our results extend the results of [3, 25] to Hadamard manifolds as well as the main result of [11] with more general assumptions on the control sequence. We also give some application to optimization.

1. INTRODUCTION Let H be a real Hilbert space with inner product , and norm . . A possibly multivalued mapping A : H −→ 2H is said to be monotone (resp. strongly monotone) operator provided that y2 −y1 , x2 −x1 0 (resp. α x1 −x2 2 ) , ∀xi ∈ D(A), ∀yi ∈ A(xi ) , i = 1, 2 , where α is a fixed positive real number and D(A) denotes the domain of A defined by D(A) := {x ∈ H : A(x) = ∅}. A is maximal monotone if and only if A is monotone and R(I + A) = H, where I is the identity mapping of H. Given any function ϕ : H −→] − ∞, +∞] (not necessarily convex) with the domain D(ϕ), its subdifferential is defined by ∂ϕ(x) := {w ∈ H | ϕ(x) − ϕ(y) w, x − y , ∀y ∈ H}. The function ϕ is called proper if and only if there exists an x ∈ H such that ϕ(x) < +∞. It is a well-known result that if ϕ is a proper, convex and lower semicontinuous function, then ∂ϕ is a maximal monotone operator. We refer the reader to the book Received March 14, 2013, accepted August 22, 2013. Communicated by Juan Enrique Martinez-Legaz. 2010 Mathematics Subject Classification: 47H05, 49J40. Key words and phrases: Proximal point algorithm, Maximal monotone operator, Resolvent, Subdifferential, Convergence, Hadamard manifold. *Corresponding author.

419

420

P. Ahmadi and H. Khatibzadeh

by Morosanu [21] in order to understand monotone operators and subdifferential of convex functions in Hilbert spaces. One of the most important problems in maximal monotone operator theory is approximation of a zero of the maximal monotone operator and one of the most popular algorithms to find a zero of a maximal monotone operator is the proximal point algorithm. This algorithm was first proposed by Martinet [19] for convex functions. The proximal point algorithm for a maximal monotone operator A : H −→ 2H , which has been introduced by Rockafellar [25], is the sequence generated by the following process (1)

yn = (1 + λn A)−1 (yn−1 + en ) ,

n = 1, 2, ...,

where {λn } is a positive real sequence and {en } is a sequence in Hilbert space H. The algorithm (1) is also a discretization of nonhomogeneous first order evolution equation of maximal monotone type. Rockafellar in his seminal paper [25] showed the weak convergence of the sequence {yn } generated by (1) to a zero of A, provided that e e zis and Lions [3] proved the weak λn λ > 0, ∀n 1, and ∞ n < +∞. Br´ n=1 2 convergence of the sequence {yn } with condition ∞ n=1 λn = +∞ on the parameter sequence {λn } and the same condition on the error sequence {en }. They also proved some other weak and strong convergence theorems with additional conditions on the maximal monotone operator A. Djafari Rouhani and Khatibzadeh [13] showed that the weak and strong convergence results of Brézis and Lions may be obtained without maximality assumption of the monotone operator A, and when the monotone operator is maximal, the weak and strong convergence point belongs to A−1 (0). In fact they proved that the weak and strong convergence theorems for the sequence {yn } are valid without assuming A−1 (0) = ∅. The second author [17] and Zaslavski [29] studied the convergence of the sequence {yn } without summability assumption on the error {en }. Convergence analysis of a modified version of the proximal point algorithm under more general error sequence studied in [5, 18, 24, 28]. Recently, the monotone operators have been defined by Németh [23] in single valued case, and by Li, Lo´ pez, Ma´ rquez and Wang [11, 12] as well as by Iwamiya and Okochi [15] in set-valued case on Hadamard manifolds. Since we aim to study multivalued monotone vector fields on Hadamard manifolds, we remind some indispensable backgrounds about Riemannian manifolds from [16] and [26]. Let M be a complete and connected m-dimensional Riemannian manifold, with a Riemannian metric , and the corresponding norm denoted by .. For p ∈ M the tangent space at p is denoted by Tp M and the tangent bundle of M by T M . Throughout the paper we assume that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature of dimension m, which is called a Hadamard manifold of dimension m.

On the Convergence of Inexact Proximal Point Algorithm on Hadamard Manifolds

421

Proposition 1.1. ([26, p. 221]). Let p ∈ M . Then expp : TpM → M is a diffeomorphism, and for any two points p, q ∈ M there exists a unique normalized geodesic joining p to q, which is, in fact, a minimal geodesic. Let [a, b] be a closed interval in R, γ : [a, b] → M a smooth curve. The length of γ is defined as b γ(t)dt ˙ L(γ) := a

and the Riemannian distance d(p, q) is defined by d(p, q) : = inf {L(γ)| γ : [0, 1] → M is a piecewise smooth curve with γ(0) = p , γ(1) = q} , which induces the original topology on M . Furthermore, d(p, q) = exp−1 p q, for any two points p, q ∈ M (see [26]). A geodesic joining p to q in M is said to be minimal if its length equals d(p, q). By definition, a geodesic triangle Δ(p1 p2 p3 ) of a Riemannian manifold is a set consisting of three points p1 , p2 and p3 , and three minimal geodesics joining these points. Proposition 1.1 shows that any m-dimensional Hadamard manifold has the same topology and differential structure as the Euclidean space Rm . In fact, Hadamard manifolds and Euclidean spaces have some similar geometrical properties. One of them is described in the following proposition. Proposition 1.2. ([26, p. 223]) (Comparison theorem for triangles). Let Δ(p1 p2 p3 ) be a geodesic triangle. Denote by γi : [0, li] → M the geodesic joining pi to pi+1 , and set li := L(γi ), αi := ∠(γ˙ i(0), −γ˙ i−1 (li−1 )), where i = 1, 2, 3 (mod 3). Then α1 + α2 + α3 π ,

(2)

2 2 − 2lili+1 cos αi+1 li−1 . li2 + li+1

Since −1 exp−1 pi+1 pi , exppi+1 pi+2 = d(pi , pi+1 )d(pi+1 , pi+2 ) cos αi+1 ,

so the inequality (2) may be rewritten as follows (3)

−1 2 d2 (pi , pi+1 ) + d2 (pi+1 , pi+2 ) − 2exp−1 pi+1 pi , exppi+1 pi+2 d (pi+2 , pi).

The monotone vector fields were first defined by Ne´ meth, [22], and the monotone point-to-set vector fields were first considered by Cruz Neto, Ferreira and Lucambio Pérez, [6]. For some important properties of monotone vector fields we refer to [7, 10]. The following definition extends some notions of the monotonicity, from the

422


corresponding notions in Hilbert spaces (see [4, 20, 21, 30]), to multivalued vector fields on Hadamard manifolds. Let X (M ) denote the set of all multivalued vector fields A : M −→ 2T M such that A(x) ⊆ Tx M for each x ∈ M and the domain D(A) of A is closed and convex, where D(A) is defined by D(A) = {x ∈ M : A(x) = ∅} . Definition 1.3. ([11]). Let A ∈ X (M ). Then A is said to be (i) monotone if the following condition holds for any x, y ∈ D(A): (4)

−1 u, exp−1 x y v, − expy x ,

∀u ∈ A(x) and

∀v ∈ A(y) ;

(ii) strongly monotone if there exists ρ > 0 such that, for any x, y ∈ D(A), we have −1 2 (5) u, exp−1 x y − v, − expy x −ρd (x, y) ,

∀u ∈ A(x) and

∀v ∈ A(y) ;

(iii) maximal monotone if it is monotone and the following implication holds for any x ∈ D(A) and u ∈ TxM : −1 (6) u, exp−1 x y v, − expy x ,

∀y ∈ D(A) and

v ∈ A(y) =⇒ u ∈ A(x).

Iwamiya and Okochi have introduced an alternative definition of monotonicity in terms of the distance function between geodesics (see [15]). It has been proved that both definitions are equivalent (see [11]). Definition 1.4. ([11]). Let A ∈ X (M ) and x0 ∈ D(A). Then A is said to be upper Koratowski semicontinuous at x0 if, for any sequences {xk } ⊆ D(A) and {uk } ⊂ T M with each uk ∈ A(xk ), the relations limk→∞xk = x0 and limk→∞ uk = u0 imply that u0 ∈ A(x0 ). A is said to be upper Koratowski semicontinuous on M if it is upper Koratowski semicontinuous at each point x0 ∈ D(A). In [11, Proposition 3.5] it has been shown that each maximal monotone vector field is upper Koratowski semicontinuous. Definition 1.5. Let X be a metric space. A map T : X → X is called nonexpansive if d(T (x), T (y)) d(x, y), for all x, y ∈ X. Definition 1.6. ([12]). Given λ > 0 and A ∈ X (M ), the resolvent of A of order λ is the set-valused mapping Jλ : M → 2M defined by (7)

Jλ (x) := {z ∈ M : x ∈ expz λAz} ,

∀x ∈ M


423

It is a well-known result that if A ∈ X (M ), D(A) = M , the vector field A is maximal monotone if and only if Jλ is single-valued and firmly nonexpansive and D(Jλ ) = M (see [12]). Ferreira and Oliveira in [9] introduced the exact (without error) proximal point algorithm (8)

0 ∈ λn A(xn+1 ) − exp−1 xn+1 xn

n = 0, 1, 2, ...,

,

where {λn} is a positive real sequence and A : M −→ 2T M is a multivalued monotone vector field. Some properties of the proximal sequence for finding singularities of vector fields were shown in [8], and recently, some important properties of the algorithm for optimization problem in Hadamard manifolds were established (see [1, 2, 27]). Li, L´ opez and M´ arquez have studied the algorithm (8) in [11] as well. The main result of [11] is the convergence of {xn } to a singularity of the monotone vector field A when λn λ > 0. It extends Rockafellar’s result when en ≡ 0 on Hadamard manifolds setting. Our aim in this paper is to extend the convergence results of Brézis and Lions in Hadamard manifolds. We study the convergence of the sequence generated by (9)

0 ∈ λn A(yn+1 ) − exp−1 yn+1 yn + en

,

n = 0, 1, 2, ...

to a singularity of the maximal monotone operator A under summability assumption on {en } and appropriate assumptions on the sequence {λn } and the maximal monotone operator A. Our results extend previous results of [3, 11, 25]. In [11] authors studied the convergence of the sequence given by (8) to a singularity of A under the assumption λn λ > 0. In this paper our motivation is to study the convergence of the sequence {yn } given by (9) to a singularity of the monotone vector field A under the more general assumptions on the control parameter {λn } and summability assumption on the error sequence {en }. Obviously, more freedom in choosing the parameters {λn } and existence the error sequence in the algorithm given by (9) can be useful from practical and computational point of views. Note that the existence of the sequence {xn } in (8) is guaranteed by the maximal monotonicity of A and Remark 4.4-(ii) of [11]. 2. CONVERGENCE RESULTS IN THE PROXIMAL POINT ALGORITHM We first recall the notion of Fejér convergence and the following related result from [14]. Definition 2.1. Let X be a complete metric space and K ⊆ X be a nonempty set. A sequence {xn } ⊂ X is called Feje´ r convergent to K if d(xn+1 , y) d(xn, y),

∀y ∈ K

and n = 0, 1, 2, ... .

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Lemma 2.2. Let X be a complete metric space and K ⊆ X be a nonempty set. er convergent to K and suppose that any cluster point of {xn } Let {xn } ⊂ X be Fej´ belongs to K. If the set of cluster points of {xn } is nonempty, then {xn } converges to a point of K. Theorem 2.3. Let A ∈ X (M ) be a maximal monotone vector field such that A−1 (0) = ∅. Suppose that x0 = y0 ∈ D(A). Assume that the sequences {xn } and {yn } are generated by the algorithms (8) and (9), respectively. If {xn } converges to a singularity of A, then {yn } does. Proof. Let {yn } converge to a singularity of A, then by (9) we have exp−1 yk yk−1 − ek ∈ λk A(yk ) ,

(10)

k = 1, 2, ... .

For every fixed k, consider the sequence {ξn (k)} defined by ξ0 (k) = yk

,

ξ1 (k) = Jλk+1 (yk ) , ... ,

ξn (k) = Jλk+n (ξn−1 (k)) .

By Theorem 4 of [12], Jλ is nonexpansive, so d(ξn (k), ξn+1 (k − 1)) = d(Jλk+n (ξn−1 (k)), Jλk+n (ξn (k − 1))) d(ξn−1 (k), ξn(k − 1)) ... d(ξ0 (k), ξ1(k − 1)) = d(yk , Jλk (yk−1 )) By definition of Jλk (yk−1 ) we have exp−1 Jλ

(11)

k

(yk−1 ) yk−1

∈ λk AJλk (yk−1 ) .

This together (4) and (10) imply that exp−1 J

λk (yk−1 )

yk−1 , exp−1 J

λk (yk−1 )

−1 yk exp−1 yk yk−1 − ek , − expyk Jλk (yk−1 )

Hence by (3), we get d(yk , Jλk (yk−1 )) ek . Therefore (12)

d(ξn (k), ξn+1 (k − 1)) ek .

By the assumption {ξn (k)} converges to some ξ(k), so (12) implies that d(ξ(k), ξ(k − 1)) ek .


425

Hence {ξ(k)} is a Cauchy sequence and so {ξ(k)} converges to some a. On the other hand d(yk , ξn+1 (k − n − 1)) d(yk , ξ1 (k−1))+d(ξ1(k−1), ξ2(k−2))+· · ·+d(ξn (k−n), ξn+1 (k−n−1)) k ei , ∀k > n , i=k−n

hence d(yn+k , ξn+1 (k − 1))

k+n

ei .

i=k

By the triangular inequality d(yk+n , a) d(yk+n , ξn+1 (k − 1)) + d(ξn+1 (k − 1), ξ(k − 1)) + d(ξ(k − 1), a). Taking limsup when n → +∞ from both sides of this inequality, we get that limsupn→+∞ d(yn , a)

+∞

ei + d(ξ(k − 1), a).

i=k

Now the theorem is proved by letting k → +∞. Theorem 2.4. Let A ∈ X (M ) be maximal monotone such that A−1 (0) = ∅. Let {λn} be a sequence of positive real numbers with (13)

+∞ n=1

λ2n = +∞ .

If y0 ∈ D(A), then the sequence {yn } generated by (9) converges to a singularity of A. Proof. By Theorem 2.3 it is enough to show the convergence of the sequence {xn }, defined by (8), to a singularity of A. For this purpose, we show that the sequence {xn } is Fejér convergent to A−1 (0) and any cluster point of {xn } belongs to A−1 (0), then one gets the result by Lemma 2.2. Let x ∈ A−1 (0). By (8), we get (14)

−1 λ−1 n+1 expxn+1 xn ∈ A(xn+1 ) ,

n = 0, 1, 2, ... .

−1 Hence the monotonicity of A, for u = λ−1 n+1 expxn+1 xn ∈ A(xn+1 ) and v = 0 ∈ A(x), implies that

(15)

−1 exp−1 xn+1 xn , expxn+1 x 0 .

426


Consider the geodesic triangle Δ(xn xn+1 x). By inequality (3) of the comparison theorem for triangles, one obtains −1 2 d2 (xn+1 , x) + d2 (xn+1 , xn) − 2exp−1 xn+1 xn , expxn+1 x d (xn , x) .

It follows from (15) that

d2 (xn+1 , x) d2 (xn , x)

er convergent to A−1 (0). which shows that {xn } is Fej´ Now, we show that any cluster point of {xn } belongs to A−1 (0). Let x be a cluster point of {xn }. Then there exists a subsequence {nk } of {n} such that xnk → x . Let −1 un := λ−1 n expxn xn−1

,

n = 1, 2, ... .

Hence un ∈ A(xn ) for each n 1 by (8). We claim that un → 0. The monotonicity of A implies that −1 un+1 , exp−1 xn+1 xn un , − expxn xn+1 ,

n = 1, 2, ... .

Hence λn+1 un+1 2 un exp−1 xn xn+1 = un exp−1 xn+1 xn = un λn+1 un+1 , which shows that the sequence {un } is nonincreasing. The inequality (3), in the geodesic triangle Δ(xn xn+1 x), and the inequality (15) show that d2 (xn+1 , xn ) d2 (xn , x) − d2 (x, xn+1 ) . Hence

λ2n+1 un+1 2 d2 (xn , x) − d2 (x, xn+1 ).

Summing up from n = 0 to n = k, and by the fact that {un } is a nonincreasing sequence, we get that (16)

uk+1

2

k n=0

λ2n+1 d2 (x0 , x) < ∞ .

By taking limit from the both sides of the inequality (16), when k → +∞, and using (13) one gets that un → 0. Thus the subsequence {unk } of {un } converges to 0 as well. Since xnk → x and A is upper Kuratowski semicontinuous at x by Proposition 3.5 of [11], 0 ∈ A(x ), that is, x ∈ A−1 (0). Now the theorem is proved by Lemma 2.2.


427

Theorem 2.5. Let A ∈ X (M ) be such that A−1 (0) = ∅. Suppose that A is maximal monotone and strongly monotone. Let {λn} be a sequence of positive real numbers such that ∞

(17)

λn = +∞ .

n=0

Let y0 ∈ D(A), then the sequence {yn } generated by (9) converges to a singularity of A. Proof. By Theorem 2.3 we only need to prove the convergence of the sequence {xn }, defined by (8), to a singularity of A. Let x ∈ A−1 (0), so 0 ∈ A(x) and −1 λ−1 n+1 expxn+1 xn ∈ A(xn+1 ) by (8). The strong monotonicity of A and (8) imply that (18)

−1 −1 2 λ−1 n expxn xn−1 , expxn x −ρd (xn , x)

n = 1, 2, ... .

Consider the geodesic triangle Δ(xn−1 xn x). By inequality (3) of the comparision theorem for triangles, we have that −1 2 d2 (xn−1 , xn) + d2 (xn , x) − 2exp−1 xn xn−1 , expxn x d (x, xn−1 ) .

It follows from (18) that 2ρλn d2 (xn , x) d2 (x, xn−1 ) − d2 (xn , x)

n = 1, 2, ...

and d2 (xn , x) d2 (xn−1 , x) n = 1, 2, ... .

(19) Hence 2ρ

k

2

λn d (xn , x)

n=1

k

(d2 (x, xn−1 ) − d2 (xn , x)) .

n=1

Combining this with (19), we obtain 2ρd2 (xk , x)

k

λn d2 (x, x◦ )

k = 1, 2, ... .

n=1

This together (17) implies that limk→∞ d2 (xk , x) = 0, that is xn → x, as n → +∞. Definition 2.6. A map A : M → 2T M is called demipositive if there exists x◦ ∈ A−1 (0) such that Ω(x◦ ) ⊂ A−1 (0), where Ω(x◦ ) is the set of all p ∈ M for which there are sequences {pn } ⊂ M and {ωn } ⊂ T M such that ωn ∈ A(pn ) , pn → p , ωn , − exp−1 pn x◦ → 0 and {ωn } is a bounded sequence.

428


Theorem 2.7. Let A ∈ X (M ) be a maximal monotone and demipositive multivalued vector field. Suppose that {λn } is a sequence of positive real numbers such that ∞

(20)

λn = +∞ .

n=1

Let y0 ∈ D(A), then the sequence {yn } generated by (9) converges to a singularity of A. Proof. By Theorem 2.3, we only need to prove that the sequence {xn }, defined by (8), is convergent to a singularity of A. Let −1 un := λ−1 n expxn xn−1

,

n = 1, 2, ... .

Hence un ∈ A(xn ) for each n 1 by (8). Since A is monotone, the same proof of er convergent to A−1 (0), and {un } is that of Theorem 2.4 shows that {xn } is Fej´ a bounded sequence. By Lemma 2.2, we need only to show that any cluster point of {xn } belongs to A−1 (0). Since A is demipositive, there exists x◦ ∈ A−1 (0) such that Ω(x◦ ) ⊂ A−1 (0). First we verify the following assertion. For any ε > 0 there exists N ∈ N such that ∀n N , ∃m ∈ N , N m n ; d(xn , xm) < ε and um , − exp−1 xm x◦ < ε . By (8), we obtain −1 −1 λk uk , exp−1 xk x◦ = expxk xk−1 , expxk x◦ ,

k = 1, 2, ... ,

and so λk uk , − exp−1 xk x◦

1 2 1 d (xk−1 , x◦ ) − d2 (xk , x◦ ) 2 2

k = 1, 2, ... .

Hence +∞

(21)

k=1

Let Since

λk uk , − exp−1 xk x◦ < ∞ .

Pε = {k ∈ N : uk , − exp−1 xk x◦ ε} . k∈Pε λk < ∞ by (21), so d(xk , xk−1 ) = exp−1 λk uk < ∞ . xk xk−1 = k∈Pε

k∈Pε

Thus there exists N1 ∈ N such that (22) kN1 , k∈Pε

k∈Pε

d(xk , xk−1 ) < ε,


429

and so there exists N N1 , by (21), such that uN , − exp−1 xN x◦ < ε . Hence for any n N , if n ∈ / Pε then assume that m = n, and if n ∈ Pε then let m be the largest integer number k < n such that k ∈ / Pε . Therefore m N and {m + 1, ..., n} ⊆ Pε , and so by (22) we obtain that d(xn , xm)

n

d(xk , xk−1 ) < ε

k=m+1

and the assertion is proved. Let x be a cluster point of A−1 (0). So there exists a subsequence {nk } of {n} such that xnk → x . By the assertion just proved, there exists a subsequence {xnj } of {xn } such that xnj → x and unj , − exp−1 xn x◦ → 0. By demipositivity of A, one j

gets that x ∈ A−1 (0) and the proof is complete by Lemma 2.2. 3. APPLICATION TO OPTIMIZATION

Recall that M is a Hadamard manifold. Let f : M → ] − ∞, +∞] be a proper, lower semicontinuous and geodesically convex function. The domain of f , D(f ) = {x ∈ M | f (x) < ∞}, is a closed convex subset of M . Consider the following minimization problem. (23)

M inM f (x)

If f is defined on a finite dimensional Hilbert space H, then M can be the constraint set of minimization f on H. Then the problem (23) can be a constraint minimization problem. The subdifferential of f at x is defined by ∂f (x) = {u ∈ TxM : u, exp−1 x y f (y) − f (x) , ∀y ∈ M }. If D(∂f ) = ∅, the subdifferential ∂f (.) is a monotone and upper Kuratowski semicontinuous multivalued vector field, and if D(f ) = M , then ∂f is a maximal monotone vector field (see Theorem 5.1 of [11]). We recall the following Lemma from [17] which is necessary to prove the following theorem. Lemma 3.1. Suppose that {an } and {bn } be two positive +∞ real sequences such } is nonincreasing and converges to zero, and that {a n=1 an bn < +∞. Then n n ( k=1 bk )an → 0 as n → +∞.

430


Theorem 3.2. Let f be a proper, lower semicontinuous, and convex function on M and A = ∂f . Let {λn } be a sequence of positive real numbers such that ∞

(24)

λn = +∞ .

n=0

Let y0 ∈ D(A), then the sequence {yn } generated by (9) converges to a singularity of A, which is a minimum point of f (by definition of f ). In addition, if en ≡ 0 then f (yn ) − f (x) = o(( ni=1 λi)−1 ), where x is a minimum point of f . Proof. By Theorem 2.3, we need only to verify that the sequence {xn }, defined by (8), is convergent to a singularity of A. For this purpose, we first prove that {xn } is Fej´ er convergent to A−1 (0). Let x ∈ A−1 (0) and n 0. Then 0 ∈ A(x) and −1 λn+1 exp−1 xn+1 xn ∈ A(xn+1 ) by (8). So the monotonicity of A implies that (25)

−1 −1 −1 λ−1 n+1 expxn+1 xn , expxn+1 x 0, − expx xn+1 = 0.

Consider the geodesic triangle Δ(xn xn+1 x). By inequality (3) of the comparision theorem for triangles, we have that −1 2 d2 (xn+1 , x) + d2 (xn+1 , xn) − 2exp−1 xn+1 xn , expxn+1 x d (xn , x) .

It follows from (25) that d2 (xn+1 , x) d2 (xn , x)

n = 0, 1, 2, ... .

Thus {xn } is Fejér convergent to A−1 (0). To complete the proof, we need only to prove that A−1 (0) contains each cluster point of {xn } by Lemma 2.2. For this purpose, we first show that {f (xn )} is a nonincreasing sequence, and f (xn ) → f (x). Since −1 λ−1 n+1 expxn+1 xn ∈ A(xn+1 ) , so by definition of the subdifferential, we have −1 −1 λ−1 n+1 expxn+1 xn , expxn+1 xn f (xn ) − f (xn+1 ) n = 0, 1, 2, ... .

Hence f (xn ) − f (xn+1 ) 0 for each n 0, and so {f (xn )} is a nonincreasing sequence. By definition of the subdifferential and the inequality (3) in the geodesic triangle Δ(xn−1 xn x), we obtain that −1 −1 f (xn ) − f (x) −λ−1 n expxn xn−1 , expxn x

1 (d2 (x, xn−1 ) − d2 (xn−1 , xn ) − d2 (xn , x)). = λ−1 2 n Hence (26)

2λn(f (xn ) − f (x)) d2 (x, xn−1 ) − d2 (xn , x),


and so 2(f (xk ) − f (x))

k

λn d2 (x◦ , x)

431

k = 1, 2, ... .

n=0

Taking limit in the previous inequality when k → +∞ and using (24), we obtain that f (xk ) → f (x). Now, let y◦ be a cluster point of {xn }, so there exists a subsequence {nk } of {n}, such that xnk → y◦ , hence by the lower semicontinuity of f , we have (27)

f (y◦ ) liminfk→∞ f (xnk ) = f (x) .

On the other hand 0 ∈ A(x) implies that x ∈ Sf , where Sf = {x ∈ M : f (x) f (y) , ∀y ∈ M } . Thus by (27), we obtain that f (y◦ ) f (y) ,

∀y ∈ M .

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P. Ahmadi and H. Khatibzadeh Department of Mathematics University of Zanjan P. O. Box 45195-313 Zanjan, Iran E-mail: [email protected] [email protected]